Problem: Complete the square to solve for $x$. $x^{2}+5x+4 = 0$
Solution: Move the constant term to the right side of the equation. $x^2 + 5x = -4$ We complete the square by taking half of the coefficient of our $x$ term, squaring it, and adding it to both sides of the equation. The coefficient of our $x$ term is $5$ , so half of it would be $\dfrac{5}{2}$ , and squaring it gives us ${\dfrac{25}{4}}$ $x^2 + 5x { + \dfrac{25}{4}} = -4 { + \dfrac{25}{4}}$ We can now rewrite the left side of the equation as a squared term. $( x + \dfrac{5}{2} )^2 = \dfrac{9}{4}$ Take the square root of both sides. $x + \dfrac{5}{2} = \pm\dfrac{3}{2}$ Isolate $x$ to find the solution(s). $x = -\dfrac{5}{2}\pm\dfrac{3}{2}$ The solutions are: $x = -1 \text{ or } x = -4$ We already found the completed square: $( x + \dfrac{5}{2} )^2 = \dfrac{9}{4}$